- Effective Medium Approximations
Effective Medium Approximations are analytical models that describe the macroscopic properties of a medium based on the properties and the relative fractions of its components. They are continuous theories and do not relate directly to percolating systems. Indeed, among the numerous effective medium approximations (EMA or EMT), only Bruggeman’s symmetrical theory is able to predict a threshold.
There are many different effective medium approximations [http://link.aip.org/link/?JAPIAU/44/3897/1 1] , each of them being more or less accurate in distinct conditions. Nevertheless, they all assume that the macroscopic system is homogeneous and generally fail to predict the properties of a multiphasic medium close to the percolation threshold due to long-range correlations.
The properties under consideration are usually the
conductivity sigma or thedielectric constant epsilon of the medium. These values are interchangeable in the formulas.Bruggeman's Model
Formulas
Circular and spherical inclusions
sum_i,delta_i,frac{sigma_i - sigma_e}{sigma_i + (n-1) sigma_e},=,0,,,,,,,,,,,,,,,,,,,,(1)
In a system of dimension n that has an arbitrary number of components [http://link.aip.org/link/?APCPCS/40/2/1 2] , the sum is made over all the constituents. delta_i and sigma_i are respectively the fraction and the conductivity of each component, and sigma_e is the conductivity of the medium. (The sum over the delta_i's is unity.)
Elliptical and ellipsoidal inclusions
frac{1}{n},deltaalpha+frac{(1-delta)(sigma_m - sigma_e)}{sigma_m + (n-1)sigma_e},=,0,,,,,,,,,,,,,,,,,,,,(2)
This is a generalization of Eq. (1) to a biphasic system with ellipsoidal inclusions of conductivity sigma into a matrix of conductivity sigma_m [http://link.aps.org/doi/10.1103/PhysRevB.18.1554 3] . The fraction of inclusions is delta and the system is n dimensional. For randomly oriented inclusions,
alpha,=,sum_{j=1}^{n},frac{sigma - sigma_e}{sigma_e + L_j(sigma - sigma_e)},,,,,,,,,,,,,,,,,,,,(3)
where the L_j's denote the appropriate doublet/triplet of depolarization factors which is governed by the ratios between the axis of the ellipse/ellipsoid. For example: in the case of a circle {L_1=1/2, L_2=1/2} and in the case of a sphere {L_1=1/3, L_2=1/3, L_3=1/3}. (The sum over the L_j 's is unity.)
Derivation
The figure illustrates a two-component medium [http://link.aip.org/link/?APCPCS/40/2/1 2] . Let us consider the cross-hatched volume of conductivity sigma_1, take it as a sphere of volume V and assume it is embedded in a uniform medium with an effective conductivity sigma_e. If the
electric field far from the inclusion is overline{E_0} then elementary considerations lead to adipole moment associated with the volumeoverline{p}, propto ,V,frac{sigma_1 - sigma_e}{sigma_1 + 2sigma_e},overline{E_0},,,,,,,,,,,,,,,,,,,,(4),,,,,,,,,,,,,,,,,,,,.
This
polarization produces a deviation from overline{E_0}. If the average deviation is to vanish, the total polarization summed over the two types of inclusion must vanish. Thusdelta_1frac{sigma_1 - sigma_e}{sigma_1 + 2sigma_e},+,delta_2frac{sigma_2 - sigma_e}{sigma_2 + 2sigma_e},=,0,,,,,,,,,,,,,,,,,,,,(5)
where delta_1 and delta_2 are respectively the volume fraction of material 1 and 2. This can be easily extended to a system of dimension n that has an arbitrary number of components. All casescan be combined to yield Eq. (1).
Eq. (1) can also be obtained by requiring the deviation in current to vanish [http://link.aps.org/doi/10.1103/PhysRevB.12.3368 4] , [http://link.aps.org/doi/10.1103/PhysRevB.13.3261 5] . It has been derived here from the assumption that the inclusions are spherical and it can be modified for shapes with other depolarization factors; leading to Eq. (2).
Modeling of percolating systems
The main approximation is that all the domains are located in an equivalent mean field.Unfortunately, it is not the case close to the percolation threshold where the system is governed by the largest cluster of conductors, which is a fractal, and long-range correlations that are totally abscent from Bruggeman's simple formula.The threshold values are in general not correctly predicted. It is 33% in the EMA, in three dimensions, farfrom the 16% expected from percolation theory and observed in experiments. However, intwo dimensions, the EMA gives a threshold of 50% and has been proven to model percolationrelatively well [http://link.aps.org/doi/10.1103/RevModPhys.45.574 6] , [http://books.google.com/books?hl=en&lr=&id=V0jr74rPdUIC&oi=fnd&pg=RA1-PR15&sig=6JFbXBYSYHyRcl6XCs38eMUblS8&dq=zallen+amorphous+solid 7] , [http://link.aip.org/link/?APPLAB/88/081902/1 8] .
Maxwell-Garnett's Equation
Formula
varepsilon_e,=,varepsilon,frac{varepsilon_m(1 + 2delta) - varepsilon(2delta - 2)}{varepsilon(2 + delta) + varepsilon_m(1 - delta)},,,,,,,,,,,,,,,,,,,,(6)
where varepsilon_e is the effective dielectric constant of the medium, varepsilon_i is the one of the inclusions and varepsilon_m is the one of the matrix; delta is the volume fraction of the embedded material.
Validity
In general terms, the Maxwell-Garnett EMA is expected to be valid at low volume fractions since it is assumed that the domains are spatially separated [http://link.aps.org/doi/10.1103/PhysRevB.74.205103 9] .
ee also
*
Percolation threshold
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